Two pairs of concentric circular coils are placed in the same plane. Pair A consists of an inner coil of radius r and an outer coil of radius R (where R r ). Pair B consists of an inner coil of radius 2r and an outer coil of radius 2R . A time-varying current I(t) with a constant rate of change flows in the outer coils of both pairs. What is the ratio of the induced emf in the inner coil of Pair A to that of Pair B?

Step-by-step solution

The induced emf in the inner coil is given by E = M dI dt , where M is the mutual inductance between the outer and inner coils. The magnetic field at the centre due to the outer coil is B = ₀ I 2R . The flux linked with the inner coil is = B r^2 = ( ₀ I 2R ) r^2 . Thus, the mutual inductance is M = I = ₀ r^2 2R . For Pair A, the mutual inductance is: M_A = ₀ r^2 2R For Pair B, the inner radius is 2r and the outer radius is 2R . The mutual inductance is: M_B = ₀ (2r)^2 2(2R) = ₀ (4r^2) 4R = ₀ r^2 R Comparing M_A and